- A stamp collector wants to include in her collection exactly one stamp from each country of Africa. If $I(s)$ means that she has stamp s in her collection, $F (s,c)$ means that stamp $s$ was issued by country $c$, the domain for s is all stamps, and the domain for $c$ is all countries of Africa, express the statement that her collection satisfies her requirement. Do not use the $\exists !$ symbol.
2.Determine whether $(p\implies q) ∧ (¬p \implies q) ≡ q$
Thank you very much for your help
I think you meant to say not to use the $\exists!$ symbol.
As for $1.$ Let $A$ denote the set of all countries in Africa and $S$ denote the set of all stamps then the proposition below $$\forall c\in A(\exists s_1\in S(I(s_1)\land F(s_1,c))\land \forall s_2\in S((I(s_2)\land F(s_2,c))\implies s_1 = s_2))$$ says that for every country $c$ in Africa there is a stamp $s_1$ in the stamp collectors collection that was issued by country $c$ and given any stamp $s_2$ in her collection that was issued by country $c$ the stamp $s_2$ must be the same as $s_1$
In short it is a claim about existence and uniqueness.
and as for $2.$ You are required to prove that $(p \implies q)\land(\neg p\implies q)$ and $q$ imply each other.
First assume that $(p \implies q)\land(\neg p\implies q)$ is true moreover we know that $p\lor \neg p$ is a tautology and in both instances when $p$ is true or false we can use our assumption in conjunction with modus ponens to deduce $q$.
For the converse notice that $p\implies q$ and $\neg p\implies q$ can be written as $\neg p\lor q$ and $p\lor q$ respectively so if you assume $q$ to be true the result follows naturally.
Hope that helps